Fix lqi_controller partial/discrete handling; add LQGProblem method; docs port

[baggepinnen]

Summary

• Fix two latent bugs in lqi_controller (src/lqg.jl):
  • Partial integrator_outputs was silently mis-wired: nr got overwritten to ny, so calling lqi_controller with anything other than the default integrator_outputs=1:ny either errored or produced a controller wired up for a different number of references than lqi had actually augmented.
  • Discrete-time plants didn't work: ss(lqi(...)) was created without a timeevol, so the gain block was always continuous-time and connect rejected it on a discrete plant with "Sampling time mismatch". Now stamps G.timeevol on it.
• Align the controller's integrator discretization with add_output_integrator(::Discrete) — both now use tf(Ts, [1, -(1-ϵ)], Ts) — and expose ϵ as a kwarg so plant augmentation and the controller agree even with leakage.
• Add lqi_controller(prob::LQGProblem, Qi; integrator_outputs, ϵ) that builds the Kalman observer internally from prob and forms Q1_aug = blkdiag(prob.Q1, Qi). This closes the f(::LQGProblem) API parity gap for lqi_controller.
• lqi docstring: replace stale Q3 in the signature with args... and clarify that the reference enters at the closed-loop level inside lqi_controller, not in the augmented plant.
• Docs port: add an "Explicit error integration via LQI" section to docs/src/lqg_disturbance.md showing the LQI alternative alongside the disturbance-model approach. This replaces the example add integrator option to extended_controller #106 was trying to add (which has been closed in favor of this approach).

Test plan

• Full test suite (Pkg.test()): 1374 pass, 6 broken (pre-existing), 0 fail.
• New test/test_lqi.jl cases:
  • MIMO continuous, all outputs integrated — reference-to-output DC gain ≈ I.
  • Partial integrator_outputs=[1] on a 2-output plant — only the integrated channel tracks exactly.
  • Discrete-time SISO with ϵ=1e-4 — verifies the integrator pole sits at 1-ϵ.
  • lqi_controller(::LQGProblem, Qi) — same dimensions and integrator behavior as the explicit-observer form.
• cd docs && julia --project make.jl — clean build; the new LQG_DIST @example block evaluates without errors.
• Closes add integrator option to extended_controller #106 (already closed remotely with a comment referencing this work).

🤖 Generated with Claude Code

[codecov-commenter]

Codecov Report

✅ All modified and coverable lines are covered by tests.
✅ Project coverage is 90.99%. Comparing base (5b9b30d) to head (ac58c12).

Additional details and impacted files

@@            Coverage Diff             @@
##           master     #139      +/-   ##
==========================================
+ Coverage   90.87%   90.99%   +0.12%     
==========================================
  Files          20       20              
  Lines        3045     3055      +10     
==========================================
+ Hits         2767     2780      +13     
+ Misses        278      275       -3     

Flag	Coverage Δ
unittests	90.99% <100.00%> (+0.12%)	⬆️

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